Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{4}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

The original coordinates of the vertices are: $R(-8,-8)$, $S(-8,8)$, $T(4,8)$, $U(4,-8)$.

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor $k$, the formula to find the new coordinates $(x',y')$ of a point $(x,y)$ is $(x',y')=(k x,k y)$. Here $k = \frac{1}{4}$.
For point $R(-8,-8)$:
$x'=\frac{1}{4}\times(-8)=-2$, $y'=\frac{1}{4}\times(-8)=-2$. So the new coordinates of $R$ are $(-2,-2)$.
For point $S(-8,8)$:
$x'=\frac{1}{4}\times(-8)=-2$, $y'=\frac{1}{4}\times8 = 2$. So the new coordinates of $S$ are $(-2,2)$.
For point $T(4,8)$:
$x'=\frac{1}{4}\times4 = 1$, $y'=\frac{1}{4}\times8=2$. So the new coordinates of $T$ are $(1,2)$.
For point $U(4,-8)$:
$x'=\frac{1}{4}\times4 = 1$, $y'=\frac{1}{4}\times(-8)=-2$. So the new coordinates of $U$ are $(1,-2)$.

Answer:

The new coordinates of the vertices are $R(-2,-2)$, $S(-2,2)$, $T(1,2)$, $U(1,-2)$